Re: Need For Speed Rivals - Start a new game
Don't own an Xbox 360 or an Xbox One, but a PS3 and assume it works in a similar way. On the PS3 there's an option to view locally stored save files and game files. Simply try and navigate around in the menus and look for save files. Delete anything related to NFS Rivals but keep the game data. Save files are usually very small (around 10KB - 2MB). There's no need to re-install the game every time.
This is a guide for the 360, so it may look a bit different on your system but it should be similar.
http://support.xbox.com/en-US/xbox-360/games/manage-games-data